Weight carried is not taken into account by WB studbook for TB stallion approval. What is taken account is what the Germans call GAG. This is called an "Official Handicap Rating" in most racing countries where Handicap races make up the meat and potatoes if everyday racing and is a rating that the Official Handicapper gives to every horse in that racing jurisdiction, from the lowliest bottom end horse, up to the best of the best. Think of it like this, if every horse in training was to line up in a Handicap race, what weight would you have to assign each horse to make it a fair contest? Obviously the best horses would carry a lot more weight than the worst horses. So G1 horses would be carrying in the 118 to 130lbs, the exceptionally good ones like Frankel and Sea the Stars probably in the mid 130s. The lowest horses, bottle of the barrel would be assigned weights down in the 40-50lb range. Obviously, you would never see horses of that disparate ability run against each other, but there are races every day of the week where horses are carrying 20lbs more than other horses. So for example, a 85 rated horse running against a 63 rated horse. In a resl life race situstion, the latter would probably be assigned 114lbs as the low weight, and the 85 rated horse would have to carry 22lbs more, so 136lbs.
There are no official handicap ratings in US races, as handicap races are rare in everyday races. Claiming races are used as the way to equalize ability... .I.e you wouldn't enter a $30k horse in a $10k claiming race and risk losing him for one third his value.
There used to be a thing in US racing called the Experimental Free Handicap, which was a similar concept, but only rated real high end horses.
I can't remember off the top of my head what the GAG rating is for TB to be approved for WB studbooks, but it's Something like the equivalent of a stakes horse a step below the graded level, like "listed" stakes level.